Peter Hart - HW 1

PHYS 360

7.3 μs

5.9 ms

euler (generic function with 1 method)

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function euler(func,x0,t)
    dt = t[2]-t[1]
    sol = zeros(length(t))
    sol[1] = x0
    for i in 1:length(t)-1
        sol[i+1] = sol[i]+func(sol[i])*dt
    end
    return sol
end

61.9 μs

Problem 2.2.2

47.7 μs

$\dot{x} = 1 - x^{14}$

$x_{f} = {- 1, 1}$

$x_{f}$	stability
1	stable
-1	unstable

6.7 μs

40.1 ms

378 ms

Problem 2.2.8

19.4 μs

$x_{f} = {- 1, 0, 2}$

$\begin{array}{lll} \frac{d \dot{x}}{d x} |_{\begin{array}{c} x = - 1 \end{array}} = 0 & \frac{d \dot{x}}{d x} |_{\begin{array}{c} x = 0 \end{array}} < 0 & \frac{d \dot{x}}{d x} |_{\begin{array}{c} x = 2 \end{array}} > 0 \end{array}$

3.4 μs

13.0 μs

$\dot{x} = (x - (- 1))^{2} (x - 0) (x - 2)$

$\dot{x} = x (x - 2) (x + 1)^{2}$

3.3 μs

53.3 ms

Problem 2.3.2

7.0 μs

$\dot{x} = k_{1} a x - k_{2} x^{2} = x \cdot (k_{1} a - k_{2} x)$

$\dot{x} (x_{f}) = 0$

$x_{f} = {0, a \frac{k_{1}}{k_{2}}}$

$\frac{d \dot{x}}{d x} = k_{1} a - k_{2} x$

$\frac{d \dot{x}}{d x} |_{\begin{matrix} x = 0 \end{matrix}} = k_{1} a > 0$

$\frac{d \dot{x}}{d x} |_{\begin{matrix} x = a \frac{k_{1}}{k_{2}} \end{matrix}} = k_{1} a - 2 k_{2} (a \frac{k_{1}}{k_{2}}) = k_{1} a - 2 k_{1} a = - k_{1} a < 0$

$x_{f}$	stability
0	unstable
$a \frac{k_{1}}{k_{2}}$	stable

2.8 μs

359 ms

Problem 2.3.3

$\dot{N} = - a N \ln (b N)$

a)

$a$ is the rate which cells grow/decay

$1 / b$ is the maximum number of cells in the tumor, as $N \to b$ the $\ln (b N) \to 0$

b)

Let $a = 1, b = 1 / 10$

9.7 μs

326 ms

Problem 2.4.5

$\dot{x} = 1 - e^{- x^{2}}$

$x_{f} = 0$

$\frac{d \dot{x}}{d x} = 2 x e^{- x^{2}}$

$\frac{d \dot{x}}{d x} |_{\begin{matrix} x = 0 \end{matrix}} = 0$

$x_{f}$	stability
0	half-stable

6.5 μs

44.6 ms

Problem 2.7.1

$\dot{x} = - \frac{d V}{d x} = x (1 - x)$

$\frac{d V}{d x} = x (x - 1) = x^{2} - x$

$\int d V = \int (x^{2} - x) d x$

$V (x) = \frac{x^{3}}{3} - \frac{x^{2}}{2} + C$

$Let C = 0$

$V (x) = \frac{x^{3}}{3} - \frac{x^{2}}{2}$

$x_{f}$	stability
0	unstable
1	stable

5.8 μs

78.8 ms

Computational Question

$\dot{x} = 1 - x^{2}$

$\frac{δ (t)}{δ (0)} = e^{λ t}$

$δ x (t) \approx e^{f^{'} (x_{0}) t}$

$λ \approx f^{'} (x_{0})$

$λ = - 2 x_{0}$

$x_{0}$	stability	$λ$
1	stable	-2
-1	unstable	2

7.1 μs

347 ms

m (generic function with 1 method)

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m(x,p) = p[1]*x  # Model to fit (linear)

22.4 μs

x
 
begin
    x01 = range(.5,stop=1.5,length=10)
    sol1 = [euler(y,x01[i],t) for i in 1:length(x01)]
    sol1_pert = [(s.-1)./(s[1]-1) for s in sol1]  # δ(x)/δ(0)
    
    plot(t,sol1_pert,lc=:blue,legend=nothing)
    xlabel!("t")
    ylabel!(L"\frac{\delta (t)}{\delta (0)}")
end

691 ms

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begin
    fits1 = zeros(length(sol1_pert))
    for i in 1:length(sol1_pert)
        fit = curve_fit(m,t,log.(sol1_pert[i]),[0.0])
        fits1[i] = fit.param[1]
    end
    λ1 = mean(fits1)
    
    plot(t,sol1_pert,yaxis=:log,lc=:blue,label=nothing)
    plot!(t,exp.(λ1 .* t),lc=:red,lw=2,label="fit")
    
    xlabel!("t")
    ylabel!(L"\frac{\delta (t)}{\delta (0)}")
end

11.2 ms

$λ_{1}$ = -1.9919314957795113

22.4 μs

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begin
    x02 = range(-.5,stop=-1.2,length=10)
    sol2 = [euler(y,x02[i],t) for i in 1:length(x02)]
    sol2_pert = [(s.+1)./(s[1]+1) for s in sol2]
    
    plot(t,sol2_pert,legend=nothing,lc=:blue)
    
    xlabel!("t")
    ylabel!(L"\frac{\delta (t)}{\delta (0)}")
end

133 ms

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begin
    fits2 = zeros(length(sol2_pert))
    for i in 1:length(sol2_pert)
        fit = curve_fit(m,t,log.(sol2_pert[i]),[0.0])
        fits2[i] = fit.param[1]
    end
    λ2 = mean(fits2)
    
    plot(t,sol2_pert,yaxis=:log,lc=:blue,label=nothing)
    plot!(t,exp.(λ2 .* t),lc=:red,lw=2,label="fit")
    
    xlabel!("t")
    ylabel!(L"\frac{\delta (t)}{\delta (0)}")
end

5.0 ms

$λ_{- 1}$ = 1.7649634965340262

6.8 μs